Perturbation Theory - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download Perturbation Theory - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! ∑ ≠ − >< = qk kq qk q EE H E )0()0( 2)0()1()0( )2( |ˆ| φψ k is not part of the degenerate set ( ) 2.0 5 1 10 2 10 2 20 10 2 1 2 1 3020 2 1|ˆ| 22 )1( 32 )1( 31 2 )0( 2 )0( 1 )1()0( 3 )0( 3 )0( 1 2)1( 31)2( 1 −=−=−= − + = − + = − >+< = − =∴ HH H EE H E ψψψ ( ) 2.0 5 1 10 2 10 2 20 10 2 1 2 1 3020 2 1|ˆ| 22 )1( 32 )1( 31 2 )0( 2 )0( 1 )1()0( 3 )0( 3 )0( 2 2)1( 32)2( 2 −=−=−= − − = − − = − >−< = − =∴ HH H EE H E ψψψ docsity.com ( ) ( ) 4.0 10 4 10 2 2 10 2 2 10 2 1 2 1 10 2 1 2 1 2030 |ˆ| 2 1 2030 |ˆ| 2 1 222 )1( 23 )1( 13 2 )1( 23 )1( 13 2 )0( 3 )1()0( 2 )0( 1 2 )0( 3 )1()0( 2 )0( 1 )0( 2 )0( 3 2)1( 23 )0( 1 )0( 3 2)1( 13)2( 3 == − += − + + = − >−< + − >+< = − + − =∴ HHHH HH EE H EE H E ψψψψψψ Summary: 4.304.0030 8.182.0120 8.202.0120 )2( 3 )1( 3 )0( 33 )2( 2 )1( 2 )0( 22 )2( 1 )1( 1 )0( 11 =++=++= =−−=++= =−+=++= EEEE EEEE EEEE docsity.com To solve this we substitute expansion into time-dependent Schrodinger Equation and solve for selected initial conditions. Reasonable choices: at t = 0 all molecules are in a particular state j. i.e. ak (o) =δkj ; that is, aj (0) =1; ak≠j (0)=0. This makes sense for electronic transitions or vibrational transitions where the Boltzmann Distributions: ( ) kT EE g e ge e N N −− ≈Excited state population Ground state population ~ 10-16 – 10-160 ~ 0 Therefore, Ng >> Ne (Note: not necessarily true for rotational energy levels at room temperature). docsity.com c) Exact Solution Preamble: dot convention for time derivatives: .;;; 3 3 2 2 etcf t ff t ff t f •••••• = ∂ ∂ = ∂ ∂ = ∂ ∂ Therefore we can write the time dependent Schrodinger wave equation as: ( ) ( ) ( )tr i trtH ,,ˆ rhr • Ω−=Ω ( )( ) ( ) ( ) ( ) ( ) t trta i trtatHH k k k k ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ −=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +⇒ ∑ ∑ ∞ = ∞ = 1 1 )1()0( , ,ˆˆ r hr φ φ ( ) ( ) ( ) ( ) ∑∑∑ ∞ = − ∞ = ∞ = ∂ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ −=+⇒ 1 )1( 1 )0( 1 )0( ˆˆ k tiE kk k k kk k k t eta i tHtaHta k h h ψ φφ Derivation ahead docsity.com